It is a remarkable theorem (capable of an `Euclidean proof', see e.g. [1]}, Thm. 4.4) that a billiard trajectory remains tangent to a confocal conic, called caustic. This will be a hyperbola or an ellipse according as the trajectory crosses or not the segment joining the foci, a property which shall be shared by all segments of trajectory. (When the trajectory passes through a focus, it will continue to pass alternatively through the two foci, which amounts to a degenerate case of the former description, when the caustic becomes the segment connecting the foci.) Let us write an affine equation for the family of confocal conics:
\(C_s:\qquad {x^2\over s}+{y^2\over s-c^2}=1,\) 
